I am trying to model a Arleigh-Burke class destroyer, but I can't make the structures above the deck symmetrical on any axis
I would suggest that you can mirror, at least long enough to get the overall shape right.  Remember that you can use a modifier for a time, and then apply it to make its effects a permanent part of your mesh.



In the N-panel you can get info about currently selected verts or groups of verts.

When you say there are structures on the ship that should be asymmetrical I would also argue that you should be splitting everything off into separate objects where possible.  You're then free to use modifiers within a more limited scope, and you can take advantage of the features of the outliner.  You can parent objects to each other to move and scale them as a group, you can use collections for hiding and showing groups of items quickly.  Depending on the level of detail you're going for here, the scene could end up very heavy anyway.  Having some level of control over your viewport performance will come in handy.
And remember this scene is your workspace right now.  These are all just organizational tools for you.  It may be that at some point you grab everything and join it together to a single object to &quot;ship it&quot;, as it were.